Morse theory on grassmanians

In the paper [N], while studying adiabatic deformations of Dirac operators on manifolds with boundary, we were led to the following nite dimensional dynamics problem. Consider (n) the grassmannian of lagrangian subspaces in the canonical symplectic space E = R2n . If A : E ! E is a selfadjoint operator anticommuting with the canonical complex structure J on E, then A belongs to the Lie algebra of the symplectic group Sp(E) and thus e is a ow of symplectic matrices. It induces a ow on (n) via the transitive action of Sp(E) on this grassmanian. This ow presents many similarities with a gradient-like ow. In particular it has a nice asymptotic behaviour. More precisely for L 2 (n) eL converges to some A-invariant lagrangian as t goes to in nity. In fact when n=1 so that (1) = S1 the phase portrait (see Fig.2 Sec.1 ) resembles the phase portrait of the gradient ow of a perfect Morse function on S1. A natural question arises. Is this ow the gradient ow (in some appropiate metric) of a Morse function on (n) ? This is one of the questions we address in these notes. The author was very pleased to nd out that this question has the best answer one can hope for. Indeed this is the gradient ow (in a natural metric) of some Morse function. This function, which must depend on A, has a simple description fA(L) = tr(APL) (0.1)